Morse coding for a Fuchsian group of a finite covolume

TL;DR

This paper investigates Morse coding of geodesic flows on hyperbolic planes with respect to Fuchsian groups, establishing a condition under which the codes form a k-step topological Markov chain based on the nature of the fundamental domain.

Contribution

It characterizes when Morse codes of geodesics form a k-step Markov chain, linking this property to the fundamental domain being an ideal polygon.

Findings

Codes form a k-step Markov chain if and only if the fundamental domain is an ideal polygon.

Provides a criterion connecting geometric properties of the fundamental domain to symbolic dynamics.

Enhances understanding of geodesic coding in hyperbolic geometry.

Abstract

We consider the Morse coding of the geodesic flow on the hyperbolic plane $H$ with respect to a Dirichlet fundamental domain $D$ of a Fuchsian group $\Gamma$. The main theorem states that the codes of all the generic geodesics constitute a $k$-step topological Markov chain, if and only if the fundamental domain $D$ is an ideal polygon (i.e. has all of its vertices on the absolute).